Introduction to Market Design | Microeconomic Analysis 1
入門マーケットデザイン | ミクロ経済分析1
Syllabus in English | 講義シラバス（日本語）
2nd semester, Mondays 2nd (10:30-12:00), Seminar Room 1（演習室1）
Office Hours: Mondays: 12:15-12:45 and 15:00-16:00
Last Update: January 5, 2020
Announcement | お知らせ
[20/1/5] --- There will be no class on January 6th.
[19/9/30] --- Please respond to the following link (here) if you plan to attend this course. <= Important!
[19/9/30] --- The first lecture will be on September 30 (Mon). Lectures will be given in English, but your questions and presentations can be made in Japanese.
Course Objective and Content | 講義概要・目的
We study economic research on auctions and matching with primary focus on market design.
This course is mainly based on students' presentations of chapters from the main textbook, Haeringer (2017), and selective academic articles.
Learning Goals | 学習目標
To acquire basic analytical skills on auctions and matching from theoretical perspectives.
To understand how theoretical ideas on auctions and matching are applied to real-life issues on market design.
Requirement / Prerequisite | 履修条件・受講条件
Game theory at intermediate undergraduate level.
Grading Policy | 成績評価
Based on the presentation and final report.
Auditing students SHOULD also make presentation but need NOT submit the final report.
Presentation can be given in English or Japanese, but SLIDES must be typed in ENGLISH.
Final report can be written in Japanese or English. Suggested length is 2 - 5 pages. Your report should be either
1) an original research paper/idea on market design,
2) a referee report on a paper (which is not directly covered in class), or
3) a survey on a specific issue of auctions, matching or market design.
Textbooks | 講義テキスト
Main Textbook
Guillaume Haeringer, Market Design: Auctions and Matching, 2017. Amazon
The author's website on the market design lecture based on the textbook is here.
Note: Slide materials on each chapter can also be downloaded :)
The relevant parts will be distributed in class, so you need NOT purchase this book.
Course Schedule | 講義日程
(Lectures 1 and 2 will be given by the instructor. All the rest consists of students' presentations.)
Lecture 1. Introduction to Market Design: Part 1 9/30
Lecture Slides
Lecture 2. Introduction to Market Design: Part 2 10/7
Lecture Slides
TEXTBOOK CHAPTERS
Lecture 3. The Basic Matching Model 10/21, 10/28
Haeringer: Chapter 9 --- Kawanaka
References
TBA
Lecture 4. The Medical Match 10/28, 11/11
Haeringer: Chapter 10 --- Fang
References
TBA
Lecture 5. Assignment Markets 11/18
Haeringer: Chapter 11 --- Lou
References
TBA
Lecture 6. Probabilistic Assignments (No Presentation)
Haeringer: Chapter 12
References
TBA
Lecture 7. School Choice 11/25
Haeringer: Chapter 13 --- Wang
References
TBA
Lecture 8. School Choice: Further Developments 12/9
Haeringer: Chapter 14 ---
References
TBA
ADVANCED TOPICS
Lecture 9. TBA 12/16
References
TBA
Lecture 10. TBA 12/23
References
TBA
Lecture 11. TBA 1/20
References
TBA
Lecture 12. TBA 1/27
References
TBA
Lecture 13. TBA 2/3
References
TBA
【以下は2016年度の講義サイトからの転載です】
講義テキスト | Textbooks
教科書: Roth and Sotomayor, Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis, 1990. Amazon (Kindle)
マッチング理論のバイブル的な教科書で、80年代までの主要な研究成果をほぼ網羅しています。
講義と関連する章はコピーを配布する予定です。
Econometric Societyの会員は（こちらからログインして）電子版を無料でダウンロードできます。
参考図書1: Roth, Who Gets What — And Why: The New Economics of Matchmaking and Market Design, 2015. Amazon (Kindle)
上の教科書の著者でもあるノーベル経済学者ロス教授による一般向けのマッチング・マーケットデザイン入門書。
日本語訳も出版されました。
参考図書2: Vulkan, Roth, and Neeman ed., Handbook of Market Design, 2013. Amazon TOC
講義に関連する章（があれば）は配布予定です。
参考図書3: Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, 1989. Amazon
計算機科学やオペレーションズ・リサーチの視点から書かれた伝統的なマッチング理論の教科書です。
サーベイ論文・書籍 | Survey Articles
Abdulkadiroglu, A. and Sonmez, T. (2013), Matching Markets: Theory and Practice, in Advances in Economics and Econometrics; 10th World Congress, Edited by Daron Acemoglu, Manuel Arellano, Eddie Dekel.
Chade, H., Eeckhout, J. and Smith, L. (2016), Sorting Through Search and Matching Models in Economics, working paper (prepared for Journal of Economic Literature)
Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences (2012), Stable Allocations and The Practice of Market Design, Scientific Background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2012. Link
Knuth, D. (1976), Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms, English Reprint (1996). Amazon
Kojima, F. (2015), Recent Developments in Matching Theory and its Practical Applications, working paper (prepared for Advances in Economics and Econometrics; 11th world congress of Econometric Society)
抄訳版が『経済セミナー』（2016年6・7月号に掲載されています）
Kojima, F. and Troyan, P. (2011), Matching and Market Design: An Introduction to Selected Topics, Japanese Economic Review, 62: 82–98.
Niederle, M., Roth, A. and Sonmez, T. (2008), Matching, in The New Palgrave Dictionary of Economics, edited by Steven N. Durlauf and Lawrence E. Blume.
Roth, A. (2002), The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics, Econometrica, 70: 1341-1378.
Roth, A. (2003), The Origins, History, and Design of the Resident Match, Journal of the American Medical Association, 289 :909-912.
Roth, A. (2008a), Deferred acceptance algorithms: history, theory, practice, and open questions, International Journal of Game Theory, 36: 537-569.
Roth, A. (2008b), What Have We Learned from Marked Design?, Economic Journal, 118: 285-310.
Sonmez, T. and Unver, U. (2011), Matching, Allocation, and Exchange of Discrete Resources, in Handbook of Social Economics, Vol. 1A, edited by J. Benhabib, A. Bisin, and M. Jackson.
Sonmez, T. and Unver, U. (2013). MARKET DESIGN FOR KIDNEY EXCHANGE. in Handbook of Market Design, 93.
Topkis, D. (1998), Supermodularity and Complementarity. Amazon (Kindle)
Vives, X. (1999), Oligopoly Pricing: Old Ideas and New Tools. Amazon
Vohra, R. (2004), Advanced Mathematical Economics. Amazon (Kindle)
小島武仁・安田洋祐, マッチング・マーケットデザイン, 経済セミナー, (647), 135-145, 2009. 増補改訂版
坂井豊貴, マーケットデザイン: 最先端の実用的な経済学, ちくま新書, 2013. Amazon
マーケットデザインの日本語による包括的な入門書です。交換問題（講義では扱わない予定）と、その応用例である臓器移植マッチングについての記述が非常に詳しいです。
安田洋祐, マッチングの数理, 数学セミナー, 4月号, 2013. 草稿
安田洋祐, 学校選択問題のマッチング理論分析, 現代経済学の潮流2014（第4章, pp.95-122）, 東洋経済新報社, 2014.
安田洋祐（編著）, 学校選択制のデザイン：ゲーム理論アプローチ（編著）, NTT出版, 2010. Amazon
学校選択制のマッチング理論分析に関する本格的な研究書です。日米の現実の制度的な背景や、マッチング理論の基礎および新展開などについて詳しく論じています。
マッチング・マーケットデザインに関する講義 | Lectures on Matching Market Design
Tayfun Sonmez | Boston College
Itay P. Fainmesser | Brown U.
Alvin E. Roth | Stanford U.
Parag Pathak | MIT
Nicole Immorlica | Northwestern U.
Yosuke Yasuda | Osaka U.
Fuhito Kojima | Stanford U.
Muriel Niederle | Stanford U.
Scott Duke Kominers | U. of Chicago
講義日程 | Lecture Schedule
Lecture Slides
Lecture 1〜4+αのスライドです。（6/10更新）
Lecture 1 One-to-One Matching Model 6/2（赤字は講義）
Two-Sided Market
Stable Matching
Gale-Shapley "Deferred Acceptance" Algorithm
M(W)-Optimal Matching
Lattice
Decomposition Lemma
Weak Pareto Optimality
Path to Stable Matching
References
**Roth and Sotomayor: Ch2
**Gale, D. and Shapley, L. (1962)
*Gale, D and Sotomayor, M. (1985a)
*Roth, A. (2008a)
Blair, C. (1988)
Demange, G., Gale, D and Sotomayor, M. (1987)
Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences (2012)
Gusfield and Irving (1989)
Knuth, D. (1976)
Roth, A. and Vande Vate, J. (1990)
Sotomayor, M. (1996)
You are expected to read all double-starred (**) articles and encouraged to read starred(*) articles.
Lecture 2 Incentive Issues 6/2
Core
Blocking Lemma
Strategy Proofness
Impossibility/Possibility Theorem
Equilibrium Behavior
Implementation
References
**Roth and Sotomayor: Ch3; Ch4
*Gale, D and Sotomayor, M. (1985b)
*Roth, A. (1982)
Alcalde, J. (1996)
Alcalde, J. and Barbera, S. (1994)
Dubins, L. and Freedman, D. (1981)
Kara, T. and Sonmez, T. (1996)
Roth, A. and Rothblum, U. (1999)
Sotomayor, M. (2004, 2007)
Lecture 3 One-to-Many Matching Model 6/9
College Admissions Model
Responsive Preferences
Strict Core
Rural Hospitals Theorem
National Resident Matching Program (NRMP)
Couples Problem
References
**Roth and Sotomayor: Ch5
*Roth, A. (1984a, 1984b)
*小島・安田 (2009)
Roth, A. and Peranson, E. (1999)
Bulow, J. and Levin, J. (2006)
Kojima, F. (2007)
Kojima, F., Pathak, P., and Roth, A. (2013)
Niederle, M. and Roth, A. (2003)
Roth, A. (1985, 1986, 1991, 2002, 2003)
Roth, A. and Sotomayor, M. (1989)
Lecture 4 Matching with Money 6/9, 16
Substitutable Preferences
Labor Market with Salaries
Gross Substitutes Condition
Salary-Adjustment Process
Budget Constraints
References
**Roth and Sotomayor: Ch6
**Kelso, A. and Crawford, V. (1982)
*Demange, G., Gale, D., and Sotomayor, M. (1986)
Crawford, V. and Knoer, E. (1981)
Demange, G. and Gale, D. (1985)
Mongell, S. and Roth, A. (1986)
Lecture 5 Matching with Contracts (1) 6/16
Contracts
Generalized Gale-Shapley Algorithm
Substitutability
Law of Aggregate Demand
Strategy-Proof Matching
Cumulative Offer Process
References
**Hatfield, J. and Milgrom, P. (2005)
*Aygun, O. and Sonmez, T. (2013)
*Echenique, F. (2012)
Lecture 6 Matching with Contracts (2) 6/16
Supply Chain
Group Incentive Compativility
Non-Substitutable Preferences
Network
Presentation
Ostrovsky, M. (2007) ---
Sonmez, T. and Switzer, T. (2013) ---
References
Kominers, S. D. (2012)
Hatfield, J. and Kojima, F. (2008, 2009, 2010)
Hatfield, J. W., and Kominers, S. D. (2015)
Westkamp, A. (2010)
Lecture 7 One-Sided Matching Model Cancel
House Allocation Problem
Serial Dictatorship
Housing Market
Top Trading Cycles
YRMH-IGYT
Presentation
Pápai, S. (2000) ---
Shapley, L. and Scarf, H. (1974) ---
References
*Sonmez, T. and Unver, U. (2011)
Roth, A. and Postlewaite, A. (1977) ---
Abdulkadiroglu, A. and Sonmez, T. (1999)
Hylland, A. and Zeckhauser, R. (1977)
Ma, J. (1994)
Pápai, S. (2003, 2007)
坂井 (2013)
Lecture 8 Kidney Exchange Cancel
Donor-Patient Pair
Blood Compatibility
Tissue Compatibility
Multi-Way Kidney Exchange
TTCC Algorithm
Two-Way Kidney Exchange
Presentation
Roth, A. E., Sönmez, T., and Ünver, M. U. (2004)
Ashlagi, I., and Roth, A. E. (2014)
References
*Sonmez, T. and Unver, U. (2013)
*Roth, A., Sonmez, T. and Unver, U. (2005, 2007)
Ashlagi, I., and Roth, A. E. (2012)
Ünver, M. U. (2010)
Lecture 9 School Choice (1) 6/30 （青字は学生報告）
One-sided Matching
Deferred Acceptance Algorithm
Boston Mechanism
Top Trading Cycles Mechanism
Acyclicity
Stable Improvement Cycles
Presentation
Abdulkadiroglu, A. and Sonmez, T. (2003) --- 小田
Ergin, H. (2002)
References
*安田 (2014)
*Kesten, O. (2010)
*Sonmez, T. and Unver, U. (2011)
Abdulkadiroglu, A., Pathak, P. and Roth, A. (2009)
Balinski M and Sonmez, T. (1999)
Chen, Y., and Kesten, O. (2013)
Erdil, A. and Ergin, H. (2008)
安田 (2010)
Lecture 10 School Choice (2) 6/30
Strategize
Naiveness
Tie-Breaking
Ex-ante Efficiency
Constrained School Choice
Preferential Treatment
Presentation
Haeringer, G., and Klijn, F. (2009) --- 方
Pathak, P. and Sonmez, T. (2013)
References
*安田 (2014)
*Abdulkadiroglu, A., Che, Y.-K. and Yasuda, Y. (2011, 2015)
*Calsamiglia, C., Haeringer, G., and Klijn, F. (2010)
Abdulkadiroglu, A., Pathak, P. and Roth, A. (2009)
Ergin, H. and Sonmez, T. (2006)
Pathak, P. and Sonmez, T. (2008)
安田 (2010)
Lecture 11 Search and Matching (1) 7/7
Frictionless Matching
Assortative Match
Assignment Game
Presentation
Chade, H., Eeckhout, J. and Smith, L. (2016): Ch1; Ch2 --- 李
References
*Becker, G. S. (1973)
*Shapley, L. and Shubik, M. (1971)
Lecture 12 Search and Matching (2) 7/7
Search Theory
Presentation
Chade, H., Eeckhout, J. and Smith, L. (2016): Ch3 --- 楊
References
...
Lecture 13 Search and Matching (3) 7/14
Search Theory
Presentation
Chade, H., Eeckhout, J. and Smith, L. (2016): Ch4 --- 呉
References
...
Lecture 14 Recent Developments 7/14
Path to Stability
Interdependent Values
Asymmetric Information
Externalities
Presentation
Chen, B., Fujishige, S., & Yang, Z. (2010) --- 塩沢
References
Klaus, B., & Klijn, F. (2007)
Kojima, F., & Ünver, M. U. (2008)
Lecture 15 Distributional Constraints 7/21
Minimum Quota
Regional Quota
Affirmative Actions
Non Substitutable Preferences
Presentation
Abdulkadiroğlu, A. (2005)
Kamada, Y. and Kojima, F. (2015)
References
*Kojima, F. (2015)
Abdulkadiroğlu, A. (2005)
Echenique, F., and Yenmez, M. B. (2012)
Ehlers, L., Hafalir, I. E., Yenmez, M. B., and Yildirim, M. A. (2014)
Goto, M., Hashimoto, N., Iwasaki, A., Kawasaki, Y., Ueda, S., Yasuda, Y. and Yokoo, M. (2014)
Goto, M., Iwasaki, A., Kawasaki, Y., Yasuda, Y. and Yokoo, M. (2015)
Kojima, F. (2012)
Kojima, F., Tamura, A., and Yokoo, M. (2014)
Hafalir, I., Yenmez, B. and Yildrim, M. (2013)
Monte, D. and Tumennasan, N. (2013)
Lecture 16 Large Market 7/21
Strategy-Proofness
Couples
Presentation
Kojima, F. and Pathak, P. (2009)
Azevedo, E. M., and Leshno, J. D. (2014)
References
*Kojima, F. (2015)
Abdulkadiroglu, A., Che, Y.-K. and Yasuda, Y. (2015)
Budish, E. (2011)
Che, Y. K., and Kojima, F. (2010)
Kojima, F., Pathak, P., and Roth, A. (2013)
Lecture 16 Recent Developments 7/21
Path to Stability
Interdependent Values
Asymmetric Information
Externalities
References
Chen, B., Fujishige, S., & Yang, Z. (2010)
Klaus, B., & Klijn, F. (2007)
Kojima, F., & Ünver, M. U. (2008)
Alternative Topic Monotone Method
Partial Order
Lattice and Sublattice
Tarski's Fixed Point Theorem
Pre-Matching
T-Mapping
Strategic Complementarities
References
**Adachi, H. (2000)
*Vives, X. (1999): Ch2
*Vohra, R. (2004): Ch7
*安田 (2013)
Echenique, F. (2005)
Echenique, F. and Oviedo, J. (2004)
Fleiner, T (2003)
Kandori, M., Kojima, F. and Yasuda, Y. (2008)
Milgrom, P. and Roberts, J. (1990, 1994)
Tarski, A. (1955)
Topkis, D. (1998)
参考論文 | References
Abdulkadiroğlu, A. (2005). College admissions with affirmative action. International Journal of Game Theory, 33(4), 535-549.
Abdulkadiroglu, A., Che, Y.-K. and Yasuda, Y. (2011), Resolving Conflicting Preferences in School Choice: the "Boston Mechanism" Reconsidered, American Economic Review, 101: 399-410.
Abdulkadiroglu, A., Che, Y.-K. and Yasuda, Y. (2015), Expanding "Choice" in School Choice, American Economic Journal: Microeconomics, 7: 1-42.
Abdulkadiroglu, A., Pathak, P. and Roth, A. (2009), Strategy-proofness vs. Efficiency in Matching with Indifferences: Redesigning the NYC Match, American Economic Review, 99: 1954-1978.
Abdulkadiroglu, A. and Sonmez, T. (2003), School choice: a mechanism design approach, American Economic Review, 93: 729-747.
Adachi, H. (2000), On a characterization of stable matchings, Economics Letters, 68: 43–49.
Alcalde, J. (1996), Implementation of Stable Solutions to Marriage Problems, Journal of Economic Theory, 69: 240–254.
Alcalde, J. and Barbera, S. (1994), Top dominance and the possibility of strategy-proof stable solutions to matching problems, Economic Theory, 4: 417-435.
Ashlagi, I., & Roth, A. E. (2012). New challenges in multihospital kidney exchange. The American Economic Review, 102(3), 354-359.
Ashlagi, I., and Roth, A. E. (2014). Free riding and participation in large scale, multi‐hospital kidney exchange. Theoretical Economics, 9(3), 817-863.
Azevedo, E. and Hatfield, J. W. (2015), Existence of Equilibrium in Large Matching Markets with Complementarities.
Azevedo, E. M., & Leshno, J. D. (2014). A supply and demand framework for two-sided matching markets. forthcoming in Journal of Political Economy.
Aygun, O. and Sonmez, T. (2013), Matching with Contracts: Comment, American Economic Review, 103: 2050-2051.
Balinski M and Sonmez, T. (1999), A tale of two mechanisms: student placement, Journal of Economic Theory, 84: 73–94.
Becker, G. S. (1973). A theory of marriage: Part I. The Journal of Political Economy, 813-846.
Blair, C. (1988), The Lattice Structure of the Set of Stable Matchings with Multiple Partners, Mathematics of Operations Research, 13: 619-628.
Budish, E. (2011). The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119(6), 1061-1103.
Bulow, J. and Levin, J. (2006), Matching and price competition, American Economic Review, 96: 652–668.
Calsamiglia, C., Haeringer, G., and Klijn, F. (2010). Constrained school choice: An experimental study. The American Economic Review, 1860-1874.
Che, Y. K., and Kojima, F. (2010). Asymptotic equivalence of probabilistic serial and random priority mechanisms. Econometrica, 1625-1672.
Chen, B., Fujishige, S., & Yang, Z. (2010). Decentralized market processes to stable job matchings with competitive salaries. KIER Discussion Paper, 749, forthcoming at Journal of Economic Theory.
Chen, Y., and Kesten, O. (2013). From boston to chinese parallel to deferred acceptance: theory and experiments on a family of school choice mechanisms.
Crawford, V. and Knoer, E. (1981), Job Matching with Heterogeneous Firms and Workers, Econometrica, 49: 437-450.
Demange, G. and Gale, D. (1985), The Strategy Structure of Two-Sided Matching Markets, Econometrica, 53: 873-888.
Demange, G., Gale, D., and Sotomayor, M. (1986). Multi-item auctions. The Journal of Political Economy, 863-872.
Demange, G., Gale, D. and Sotomayor, M. (1987), A further note on the stable matching problem, Discrete Applied Mathematics, 16: 217–222.
Dubins, L. and Freedman, D. (1981), Machiavelli and the Gale-Shapley algorithm, American Mathematics Monthly, 88: 485–494/
Echenique, F. (2005), A Short And Constructive Proof of Tarski's Fixed-Point Theorem, International Journal of Game Theory, 33: 215-218.
Echenique, F. (2012), Contracts vs. Salaries in Matching, American Economic Review, 102: 594-601.
Echenique, F. and Oviedo, J. (2004), Core many-to-one matchings by fixed point methods, Journal of Economic Theory, 115: 358–376.
Echenique, F., and Yenmez, M. B. (2012). How to control controlled school choice. Available at SSRN 2189706.
Erdil, A. and Ergin, H. (2008), What's the Matter with Tie-Breaking? Improving Efficiency in School Choice, American Economic Review, 98: 669-689.
Ergin, H. (2002), Efficient resource allocation on the basis of priorities, Econometrica, 70: 2489–2497.
Ergin, H. and Sonmez, T. (2006), Games of school choice under the Boston mechanism, Journal of Public Economics, 90: 215–237.
Ehlers, L., Hafalir, I. E., Yenmez, M. B., and Yildirim, M. A. (2014). School choice with controlled choice constraints: Hard bounds versus soft bounds. Journal of Economic Theory, 153, 648-683.
Fleiner, T (2003), A fixed-point approach to stable matchings and some applications, Mathematics of Operations Research, 28: 103–126.
Gale, D. and Shapley, L. (1962), College Admissions and the Stability of Marriage, American Mathematical Monthly, 69: 9-15.
Gale, D and Sotomayor, M. (1985a), Some remarks on the stable matching problem, Discrete Applied Mathematics, 11: 223-32.
Gale, D and Sotomayor, M. (1985b), Ms Machiavelli and the stable matching problem, American Mathematical Monthly, 92: 261-8.
Goto, M., Hashimoto, N., Iwasaki, A., Kawasaki, Y., Ueda, S., Yasuda, Y. and Yokoo, M. (2014), Strategy-proof Matching with Regional Minimum Quotas, AAMAS2014 (Full Paper), 1225-1232.
Goto, M., Iwasaki, A., Kawasaki, Y., Yasuda, Y. and Yokoo, M. (2015), Improving Fairness and Efficiency in Matching with Distributional Constraints: An Alternative Solution for the Japanese Medical Residency Match..
Haeringer, G., and Klijn, F. (2009). Constrained school choice. Journal of Economic Theory, 144(5), 1921-1947.
Hafalir, I., Yenmez, B. and Yildrim, M. (2013), Effective Affirmative Action in School Choice, Theoretical Economics, 8: 325-363.
Hatfield, J. W., and Kojima, F. (2008). Matching with contracts: Comment. The American Economic Review, 98(3), 1189-1194.
Hatfield, J. W., and Kojima, F. (2009). Group incentive compatibility for matching with contracts. Games and Economic Behavior, 67(2), 745-749.
Hatfield, J. W., and Kojima, F. (2010). Substitutes and stability for matching with contracts. Journal of Economic Theory, 145(5), 1704-1723.
Hatfield, J. W., and Kominers, S. D. (2015). Multilateral matching. Journal of Economic Theory, 156, 175-206.
Hatfield, J. and Milgrom, P. (2005), Matching with Contracts, American Economic Review, 95: 913-935.
Hylland, A. and Zeckhauser, R. (1977), The efficient allocation of individuals to positions, Journal of Political Economy, 87: 293-314.
Kamada, Y. and Kojima, F. (2015), Efficient matching under distributional constraints: Theory and applications, The American Economic Review, 105(1), 67-99.
Kara, T. and Sonmez, T. (1996), Nash Implementation of Matching Rules, Journal of Economic Theory, 68: 425–439.
Kandori, M., Kojima, F., and Yasuda, Y. (2008), Understanding Stable Matchings: A Non-Cooperative Approach, mimeo.
Kelso, A. and Crawford, V. (1982), Job Matching, Coalition Formation, and Gross Substitutes, Econometrica, 50: 1483-1504.
Kesten, O. (2010), School Choice with Consent, Quarterly Journal of Economics, 125: 1297-1348.
Klaus, B., & Klijn, F. (2007). Paths to stability for matching markets with couples. Games and Economic Behavior, 58(1), 154-171.
Kojima, F. (2007). Matching and price competition: comment. The American economic review, 1027-1031.
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